A Mathematical Model for the Analysis of the Pressure Transient Response of Fluid Flow in Fractal Reservoir

This study uses similar construction method of solution (SCMS) to solve mathematical models of fluid spherical flow in a fractal reservoir which can avoid the complicated mathematical deduction. The models are presented in three kinds of outer boundary conditions (infinite, constant pressure, and closed). The influence of wellbore storage effect, skin factor, and variable flow rate production is also involved in the inner boundary conditions. The analytical solutions are constructed in the Laplace space and presented in a pattern with one continued fraction—the similar structure of solution.The pattern can bring convenience to well test analysis programming.Themathematical beauty of fractal is that the infinite complexity is formed with relatively simple equations. So the relation of reservoir parameters (wellbore storage effect, the skin factor, fractal dimension, and conductivity index), the formation pressure, and the wellbore pressure can be learnt easily. Type curves of the wellbore pressure and pressure derivative are plotted and analyzed in real domain using the Stehfest numerical invention algorithm. The SCMS and type curves can interpret intuitively transient pressure response of fractal spherical flow reservoir. The results obtained in this study have both theoretical and practical significance in evaluating fluid flow in such a fractal reservoir and embody the convenience of the SCMS.


Introduction
The mechanics of oil and gas seepage is a discipline which researches the law and state of fluid flow in porous media.The practical development of oil and gas reservoirs shows that reservoir distribution and its space structure are awfully complicated.In 1982, Mandelbrot and Blumen first proposed the fractal geometry theory which used self-similar to characterize the complexity of things [1].By combining the theories of fractal and seepage mechanics, it is able to describe fluid flow paths in porous media effectively.Acuna et al. explained the fractal characters of natural porous media with extremely complex pore structure [2].Because flow paths of fluid in porous media can be seen as tortuous capillaries, Cai and Yu applied a fractal dimension to describe the tangle of capillary pathways and derived calculation formula of actual length of tortuous flow paths [3].And spontaneous imbibition of wetting liquid in fractal porous media including gravity was studied [4].The researches were regarded as a crucially important driving mechanism for enhancing oil recovery in naturally fractured reservoir.In the early 1990s, Chang and Yortsos applied the fractal theory in reservoir models and then set up new mathematical models [5,6].Tian and Tong established radial fluid flow models of fractal reservoirs [7].The models have shown that the order of the fractional dimension has influence on the whole pressure behavior.Particularly, the effect on pressure behavior is larger in the early time stage.Based on fractal geometry and the semiempirical Kozeny-Carman equation which is the most famous permeability-porosity relation in the field of flow in porous media, Xu and Yu [8] developed a new form of permeability and Kozeny-Carman constant.After that, they showed that fractal dimension had significant effect on permeability, which can enhance the effective permeability [9,10].Santizo discussed the effect of fractal dimension and fractal conductivity index on pressure derivative in finite-conductivity fractures [11].The papers obtained some numerical or analytical solutions of the formation pressure and the wellbore pressure in different fractal reservoirs or plotted pressure-time curves to analyze the influence of reservoir parameters [12][13][14][15][16][17].
Different versions of cylindrical flow model have been studied on the assumption that wells were completely opened and fully penetrated the productive formation.However, the assumption may seem too restrictive for practical application.Reservoirs have vertical permeability and the length of injection or extraction region was small compared to thick formations; spherical flow model can provide a good approximation in practice [18].Schroth and Istok [19] illustrated the applicability of the derived spherical flow solution and provided a comparison with its cylindrical flow counterpart.They showed the spherical flow solution increased with increasing anisotropy in hydraulic conductivities.Joseph and Koederitz [20] presented short-time interpretation methods for radial-spherical flow in homogeneous and isotropic reservoirs inclusive of wellbore storage, wellbore phase redistribution, and damage skin effects.Liu [21] studied the transient spherical flow behavior in porous media and derived its nonlinear partial differential equation and obtained its analytical, asymptotic, and approximate solutions by using the methods of Laplace transform.
In the above proposed studies, the solution processes of their mathematical models are very complicated.Based on some study in a boundary value problem of the secondorder linear homogeneous differential equation, Li and Liao recently introduced a new idea, similar construction method.The similar structure theory and its application were developed maturely [22].Chen and Li presented similar construction method of solution (SCMS) for the boundary value problem of Bessel equation and gave the method to analyze the structure characteristics of solution [23].After that, Sheng et al. proposed SCMS for a fractal reservoir and listed its steps [17].They verified that the SCMS is a straightforward method to solve mathematical models in reservoir engineering, but they just did a theoretical study.SCMS of the boundary value problems of Airy equation, Weber equation, Euler hypergeometric equation, and composite first Weber system were studied [24][25][26][27][28]. SCMS in fractal dual-porosity reservoir and dual-permeability reservoir were presented [29][30][31].
This paper presents a mathematical model for the analysis of the pressure transient response of fluid spherical flow in fractal reservoir, which considers wellbore storage and skin effect in inner boundary conditions.According to a mathematical method, called SCMS, the expressions of the dimensionless formation pressure and the dimensionless wellbore pressure are constructed in the Laplace space, which avoid complicated calculations.Type curves of the wellbore pressure and pressure derivative responses are plotted and analyzed in real domain using the Stehfest numerical invention algorithm [32].The results obtained in this study have essential significance to understand the pressure characteristics and provide theoretical basis in such a reservoir.This paper embodies the relationship between engineering and mathematics.

Reservoir Characteristics.
Nonpenetrating wells that occur in a thick formation can be treated as spherical systems, as shown in Figure 1.Using fractal theory to describe the pore nature and transport properties, we assume that dimensional fractal network   is embedded in the dimensional Euclidean rock .The fluid only flows in the fractal network, and the flow will obey Darcy's law from the reservoir into the wellbore.The reservoir has a uniform thickness of ℎ and original formation pressure is   .The fractal reservoir is mined with a single small opening hole in the top part.The fluid flow rate is ().The gravity and the capillary pressure are ignored.

Dimensionless Mathematical Model. The dimensionless mathematical model of the reservoir is made up of four parts, such as continuity equation, initial condition, inner boundary condition, and outer boundary condition. Appendix A presents mathematical model of fluid flow and dimensionless variables.
The continuity equation of fluid spherical flow in a fractal reservoir is Initial condition: Inner boundary condition: Three kinds of outer boundary conditions are the following cases.
Case 1. Infinite outer boundary condition: Case 2. Constant pressure outer boundary condition: Case 3. Closed outer boundary condition:

A Boundary Value Problem of the Modified Bessel Equation.
The dimensionless mathematical model (( 1)-( 6)) happens to be a boundary value problem of a second-order partial differential equation.It can be transformed into a boundary value problem of a second-order ordinary differential equation based on Laplace transform and then it can be further simplified as a boundary value problem of the modified Bessel equation:

Solution.
The SCMS is based on a boundary value problem of a second-order linear differential equation and is firstly proposed by Sheng et al. [17].Using coefficients of its right and left boundary condition and its two linearly independent solutions of the differential equation, the solution of the boundary value problem can be obtained.According to the steps of SCMS, the solution of the dimensionless mathematical model of fractal spherical flow reservoir can be constructed.The details are as follows.
Step 2. Construct a binary function  0,0 ( 1 ,  2 ) employing the linearly independent solutions (10) and calculate its partial derivatives for  1 ,  2 , respectively: where with real numbers , : Step 3. Construct similar kernel functions Φ(  , ) using the outer boundary conditions ( 9), the binary function equation (11), and its partial derivatives equations (13): Step 4. Construct the similar structure of solution with the similar kernel function equation ( 14) and coefficients (  (),   , ) of the inner boundary condition equation ( 8): Equation ( 15) is the analytical solution of the mathematical model of fluid spherical flow in a fractal reservoir.Substitute (15) into the second expression of (3); the dimensionless wellbore pressure in the Laplace space can be obtained:

Results and Discussions
The analytical solution of the mathematical model of fluid flow in fractal reservoir was rewritten as continued fraction, like real numbers, the structure of continued fraction is very beautiful.It is very convenient to study the influence of both wellbore storage effect and skin factor using equation (15).For different outer boundary conditions, the solution has a different kernel function, but its expression is the same so that it is convenient to program well test analysis software.Both the dimensionless formation pressure and the dimensionless wellbore pressure in the Laplace space can be inverted back into real time space using the Stehfest numerical invention algorithm (the numerical invention details are shown in Appendix B).The analysis of pressure transient response of fluid flow in fractal reservoir is studied by type curves.

Pressure Transient Response.
Figure 2 shows the dimensionless wellbore pressure and derivative responses in loglog coordinate for three kinds of outer boundary conditions (infinite, constant pressure, and closed), which can be divided into the following four seepage flow stages.Stage 1 (wellbore storage period).The fluid flow is principally affected by wellbore storage effect.In this flow period, both the dimensionless pressure change and its rate are aligned in a unite slope trend.
Stage 2 (transition flow period).The period represents the transition to early time spherical flow (Stage 3) and has a hump in the dimensionless pressure derivative curve.The fluid flow depends on both the wellbore storage effect and the skin factor.
Stage 3 (spherical flow before the outer boundary period).
During this period, either the dimensionless pressure change curve or the derivative curves show a straight line with a slope of zero.The greater the radius of the outer boundary is, the longer the duration of the fluid flow period will be.

Stage 4 (spherical flow after the outer boundary period).
When the outer boundary is infinite, the fluid flow still holds the same response.When pressure is constant in the outer boundary, the dimensionless pressure curve nearly remains level, but its derivative curve decreases quickly.For a closed outer boundary condition, both the dimensionless pressure and derivative curves are cocked up and overlapped.And the values of their slope curve are about 1.
As Figure 2 shows, the differences of both the dimensionless wellbore pressure and derivative responses in various outer boundary conditions only display in Stage 4.
For convenience, we only analyze infinite outer boundary condition.The effect of reservoir parameters trends to be similar among the three kinds of outer boundary conditions (infinite, constant pressure, and closed), so we analyze the case of infinite outer boundary condition in the following.

Effect of Wellbore Storage.
Figure 3 shows the effect of the coefficient of wellbore storage,   , on pressure transient responses of a fractal reservoir with spherical flow.It can be seen that, with the other parameters kept constant, the curves have obvious impact on the wellbore storage period and the   transition flow period.The curves of dimensionless pressure derivative indicate that the larger coefficient of wellbore storage increases the duration of the wellbore storage period.

Effect of Skin Factor.
Figure 4 shows the effect of the skin factor, , on pressure transient responses of a fractal reservoir with spherical flow.If the other parameters keep constant, the larger the skin factor is, the higher the hump of the wellbore storage period in the derivative becomes.

Effect of Fractal Dimension.
The effect of the fractal dimension,   , is shown in Figure 5.We can find that the fluid flow behaviors are not affected in the wellbore storage period.The bigger the   is, the higher the derivative curve will become.As time goes on, the difference could be observed obviously.That is because of the fact that, with fractal dimension increasing, the fractal porosity and  permeability of a reservoir will become bigger, which also can be understood from (A.1).

Effect of Conductivity Index.
Figure 6 shows the influence of conductivity index, , on pressure transient responses of a fractal reservoir with spherical flow.The conductivity index reflects the curvature of the pore system.It holds that the more circuitous pore system is, the worse connectivity and conductivity will be.When the fractal dimension is small enough and makes  +  −   < 1, the derivative curve decreases at the late time such as the blue curve and the green curve because the reservoir is similar to the situation of the constant pressure outer boundary.When the fractal dimension makes  +  −   ≥ 1, the derivative curve (the red curve) inclines upward at the late time because the reservoir is similar to the situation of the closed outer boundary.

Conclusions
Based on the reasoning mentioned above, the following conclusions can be drawn.
(1) The mathematical models of fluid spherical flow in fractal reservoirs with three kinds of outer boundary conditions (infinite, constant pressure, and closed) were presented, which comprehensively took into consideration the effect of wellbore storage effect, the skin factor, fractal dimension, and conductivity index.(2) The dimensionless formation pressure in the Laplace space was constructed by SCMS and was written as a continued fraction.The analytical solution presented in this paper has afforded theoretical basis to understand the effect of reservoir parameters and practical significance to make the well test analysis software.
(3) The dimensionless pressure and derivative type curves were plotted and their characteristics were analyzed.For three kinds of outer boundary conditions, the pressure transient responses make no difference when the fluid does not reach to the outer boundary condition.(4) The time and the high of the hump of the dimensionless wellbore pressure derivative have connection with the coefficient of wellbore storage   and the skin factor , respectively.The paper has studied the effects of the conductivity index  and the fractal dimension   .The fractal dimension   is beneficial to the fluid flow.However, the trend is reversing with the conductivity index .In addition, the behaviors in wellbore storage period are not affected for the fractal dimension   and the conductivity index .

A. Mathematical Model of Fractal Spherical Flow Reservoir
In Acuna et al' .sresearch [2], the fractal porosity and permeability of a reservoir as a function of distance are where ,   , , and  represent Euclid dimension, fractal dimension, conductivity index, and radial distance, respectively.
The fluid flow from reservoir into the wellbore follows Darcy's law.The fluid flow in the reservoir is isothermal and single-phase with constant fluid viscosity.Based on the principle of mass and energy conservation, the mathematical equations of single fluid phase in fractal reservoir can be written in the form as  (A.5) The three kinds of outer boundary condition are as follows.For the convenience of calculation, the dimensionless variables (formation pressure, wellbore pressure, production rate, radial distance, outer boundary radius, time, and wellbore storage coefficient) are defined as follows: (A.9) Taking the above dimensionless variables into (A.2)-(A.8),the dimensionless mathematical model of fractal reservoir ((1)-( 6)) can be obtained.

B. Stehfest Numerical Invention Algorithm
The solution in real space can be obtained by the method of the Stehfest numerical invention algorithm [32].  (  ) is the primary function of   () using the Laplace transform; then   (  ) is given by Wooden et al. [33].Consider Here  is an even number which is usually in the range 10 <  < 30, and  is computed using integer arithmetic.

Figure 1 :
Figure 1: Schematic for spherical flow in a reservoir.

Figure 2 :
Figure 2: The type curves of the fractal reservoir with spherical flow.

Figure 3 :
Figure 3: Effect of   for the type curves of the fractal reservoir with spherical flow.

Figure 4 :
Figure 4: Effect of  for the type curves of the fractal reservoir with spherical flow.

Figure 5 :Figure 6 :
Figure 5: Effect of   for the type curves of the fractal reservoir with spherical flow.